= Isobaric recombination-cooling stability with constant volumetric heating
{title2=$\mathcal L_T=(\Gamma/T_0)(d\ln\alpha_B/d\ln T-1)$}
Take fully ionized <hydrogen> with electron and proton <number densities> both $n$, fixed local heat input $\Gamma$ per unit volume, and volumetric cooling $3n^2\alpha_B(T)k_BT/2$. The <ideal gas law> is $P=2nk_BT$. At fixed <pressure>, net cooling becomes $\mathcal L=3P^2\alpha_B(T)/(8k_BT)-\Gamma$. At an equilibrium $T_0$, differentiation gives the displayed derivative.
The linearized thermal equation is $\rho_0c_p\dot{\delta T}=-\mathcal L_T\delta T$, so equilibrium is stable exactly when the logarithmic slope of $\alpha_B$ exceeds one. A constant or decreasing coefficient gives instability. A different heating prescription must be differentiated too; constant incident radiation does not generally imply a density-independent local heating rate.
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